<p>In this paper, we construct an approximation of the fundamental solution of a problem for a hyperbolic system of first-order linear differential equations with constant coefficients. We propose an algorithm for the approximate solution of the generalized Riemann problem on the decay of a discontinuity under additional conditions on the boundaries. This algorithm reduces the problem of finding values of variables on both sides of the discontinuity surface of the initial data to solving a system of algebraic equations whose right-hand sides depend on the values of the variables at the initial moment of time at a finite number of points. Based on these solutions, we develop a computational algorithm for the approximate solution of the initial-boundary-value problem for a hyperbolic system of first-order linear differential equations. The algorithm is implemented for a system of equations of elastic dynamics; moreover, we use it to solve some applied problems related to oil production.</p>